Question: Multiply the following complex numbers, marked as blue dots on the graph: $(7 e^{\pi i}) \cdot ( e^{\pi i / 4})$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $7 e^{\pi i}$ ) has angle $\pi$ and radius $7$ The second number ( $ e^{\pi i / 4}$ ) has angle $\frac{1}{4}\pi$ and radius $1$ The radius of the result will be $7 \cdot 1$ , which is $7$ The angle of the result is $\pi + \frac{1}{4}\pi = \frac{5}{4}\pi$ The radius of the result is $7$ and the angle of the result is $\frac{5}{4}\pi$.